(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.

Q: Q6: ) Suppose (X, 7) is aftopological space, and p a point not in X. Let Q = {UU {p}, UET, and X-U…

A: Let's go through the two parts of the question:a) To show that T=Q∪T is a topology on Y=X∪{p}, need…

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: 2. Let (X,r) be a topological space. Where X {a,b.c,d,e} and…

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: if a family iP; Sierof sominorms on avectorspace X separates distinct points of x , show that the…

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Q: Tx) and (X, Ty) be two topological spaces. Let X-{a, b, c, d}. Tx-{4, X, {a}, {a, b}, {a, b, c}} Y…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (3) Consider the topological space (R. 7), where T= {UCR: U = 0 or R\U is countable}. (a) Show that…

A: As we know τ is a topology on R if1. Both the empty set and R are elements of τ.2. Any union of…

Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

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Q: Let c[o] be with metrics the Space d, (f, g) - max f(x)-gen) fen) = x2 4 gcx) =x The find d, Cf,g)…

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Q: * Let R be with the indiscrete topology. If A = {1,3, 5, 7, . }, then A' = A O N O RO

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

Q: A ° = A – JA

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

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Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: d T = {0, {a}, {a, b}, {c,d}, {a, c, d}, {a,b, c, d}, X}. (1) Is this space To? Why or why not? (2)…

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Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

A: Given: X is a topological space. To Prove: We have to show that the topological space X itself is a…

Q: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

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Q: d set): Let (M ,d) be a (SUT)' = S' U T'.

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Q: 5. Let X =[0,2). Define T = {[0,a):0<a<2}. %3D a) Show that T is a topology on X. b) What are the…

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Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

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Q: (i) Show that a subset A of a topological space X is closed if and only A = A (ii) Prove that A…

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Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

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Q: 5) Let X be a topological space. Is there always a metric on X which gives rise to its topology?…

A: No, there does not always exists a metric in a topological space that gives rise to its topology.…

Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: Let (IR, I) be usual 2 2 IR ² X IR² = Cx C space. Prove

A: Solution:  The  given equation f:ℝ2→ℂ   defined  by  f(a,b)=(a+ib), a,b∈ℝ2. Then,a,b∈ℝ2, and…

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

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Q: Exercise 2. Let X be a topological space. Endow X X X with the product topology. Consider the map f:…

A: We have to show for every closed subset C , f(C) is closed in X×X

Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: A of a metric space (x, d.)

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Q: Q2. Answer as true or false: (a) f-'((AUB)nC) = (ƒ-'(A)n f-1C)) U (f-1(B) n f-1(C)). (b) Given the…

A: (a) is True. (b) is False.

Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: ANB +P

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